Triple integration, Spherical coordinates

Question

How do we get limit such as $0\le\theta\le\pi$, $0\le\phi\le2\pi$ in spherical coordinate system where $$x=r \sin\theta\cos\phi, y=r \sin\theta\sin\phi, z=r \cos\theta$$ Why is the $\theta$-limit $[0,\pi]$ and not $2 \pi$? I think that the angle of the foot of perpendicular from the point taken in spherical coordinate can lie within $[0,2 \pi]$

Answer 1

You will be counting points more than once if you choose your angle like that (just take a sphere and walk over it using your angles and you will see what I mean). For example, \begin{align} (0,0,r)&=(r\sin(0)\cos(0),r\sin(0)\sin(0),r\cos(0))\\ &=(r\sin(2\pi)\sin(0),r\sin(2\pi)\cos(0),r\cos(2\pi)). \end{align}

Answer 2

$\theta$ is the angular distance from the north pole of the sphere. It shouldn't be more than $\pi$, because once you've gone past the south pole, you should be measuring along the opposite meridian instead (that is, with $\phi\pm\pi$.